Note: I'm not in a position to do any of the math right now, so I'm more addressing the "wondering if there are any ways to reduce the amount of Delta-V used" part of the question.
Gravity assists can go both ways (acceleration or deceleration) depending on the geometry of the approach. Note that entering orbit using a gravity assist probably requires you to drop into a retrograde orbit, which may be undesirable depending on what you're trying to accomplish. For example, if you mean to intercept a moon, a gravity assist would be counterproductive because you'd wind up going the wrong way around the planet and approaching all the moons head-on. But we believe that's how some planets ended up with retrograde moons -- they just happened to fall in on a trajectory where the gravity assist slowed them into a stable orbit going the "wrong" way.
Another trick is the Oberth effect. For some complicated physics reasons, fuel becomes more efficient the deeper you are in a gravity well. This doesn't change the delta-V required, but it increases how much delta-V a given mass of fuel is worth. For example, Jupiter has an incredibly huge gravity well, so thrusters are extremely efficient when used during a close, high speed pass. That efficiency counts whether you're using the thrusters to speed up or slow down, so aiming for a close initial approach is a good strategy even if your target orbit is much further out. The price you pay is that close approach will be very fast, which means you don't stay there for very long, so you need to use thrusters that have a very high impulse. In other words, Oberth makes it more efficient if you can blast out a lot of fuel very fast, but highly efficient thrusters usually throw out a very small amount of fuel at a time, so there's a balancing act there to get your approach as close as possible while still staying close long enough to fire the thrusters for the necessary time. (If it's Jupiter, you also run into issues where the space environment gets significantly more dangerous in terms of radiation/plasma fields the closer you get to the planet, so you have to also consider how that's going to impact your craft.)
Aerobraking is also an option, and in theory you can bleed off any arbitrary amount of velocity that way, given a sufficiently thick atmosphere. However, it's a surprisingly tricky problem, especially with very large planets, because tiny changes in the approach angle can have a huge effect on the distance traveled through the atmosphere. There are other issues with aerobraking, depending on how aggressively you're going to do it, like shaping the craft and adding thermal shielding to ensure you don't damage anything in the process (which may end up making aerobraking less effective than just loading a bit more fuel). Actual aerocapture is probably so tricky that it isn't worth the effort as compared to aerobraking down to a reasonable speed and using thrusters to finish the orbital capture.
